October 24, 2018 9:11 Proceedings Trim Size: 9in x 6in des SOME APPLICATIONS OF THE ADM FORMALISM∗ J. E. NELSON Dipartimento di Fisica Teorica, Universit`adegli Studi di Torino and Istituto Nazionale di Fisica Nucleare, Sezione di Torino via Pietro Giuria 1, 10125 Torino, Italy E-mail: [email protected] The ADM Formalism is discussed in the context of 2 + 1–dimensional gravity, uniting two areas of relativity theory in which Stanley Deser has been particularly active. For spacetimes with topology IR×T 2 the partially reduced and fully reduced ADM formalism are related and quantized, and the role of ”large diffeomorphisms” (the modular group) in the quantum theory is illustrated. 1. Introduction Over forty years ago Arnowitt, Deser and Misner (ADM) studied the 3+1– decomposition of general relativity, its initial value problem, the dynamical structure of the field equations and calculated the Hamiltonian 1. This extraordinary piece of work has become a fundamental ingredient of modern arXiv:gr-qc/0408083v1 25 Aug 2004 relativity theory. It is now regularly taught as an integral part of relativity courses, and usually occupies at least a chapter in relativity textbooks. The problem had actually been considered previously by Dirac 2 who applied his theory of constrained systems 3 to the gravitational field. But Dirac’s treatment was incomplete and in a particular gauge. In this Section I briefly summarise the ADM results,and in Section 2 discuss the main differences between the 3 and 4 dimensional theories. In Section 3 the second–order, partially reduced, ADM formalism, for space- times of topology IR T 2 is reviewed, and I show how in principle the × system can be quantized. In Section 4 the first–order fully reduced holon- omy approach is presented. In Section 5 the two approaches are related, both classically and quantum mechanically, using the action of the modular group, or “large diffeomorphisms - those that remain after ADM reduction. ∗Work supported by the Istituto Nazionale di Fisica Nucleare (INFN) of Italy and the Italian Ministero dell’Universit`ae della Ricerca Scientifica e Tecnologica (MIUR). 1 October 24, 2018 9:11 Proceedings Trim Size: 9in x 6in des 2 The 3 + 1–decomposition of the Einstein–Hilbert action calculated by ADM is 4 (4) (4) 3 ij i IEin = d x g R = dt d x π g˙ij N i N , (1) Z − Z Z − H − H p Σ where spacetime is of the form IR Σ, and time runs along IR. In (1) the × metric has been decomposed as ds2 = N 2dt2 g (dxi + N idt)(dxj + N j dt) i, j =1, 2, 3 (2) − ij and πij = √g (Kij gij K), where Kij is the extrinsic curvature of the − surface Σ labeled by t = const.a In (2) the lapse N i and shift N functions are related to the non-dynamical components of gij and their variation in (1) leads to the supermomentum and super-Hamiltonian constraints on gij and πkl. j 1 ik jl 1 ij kl i = 2 jπ i =0, = gij gkl(π π π π ) √gR = 0 (3) H − ∇ H √g − 2 − where is the covariant derivative for the connection compatible with ∇j gij , and indices are now raised and lowered with gij . The , i in (3) kl 1 H H are non-polynomial in gij and π and involve (gij )− . They are directly proportional to the components Goµ of the Einstein tensor defined by δIEin 1 Gµν = = Rµν Rgµν (4) δgµν − 2 so finding a solution to (3) would correspond to finding a general solution of Einstein’s equations (the other components of (4) are zero by the Bianchi µν identities G ν = 0). | The constraints (3) generate, through the Poisson brackets obtained from (1) 1 g (x), πkl(y) = (δkδl + δkδl)δ3(x y) (5) { ij } 2 i j j i − three-dimensional diffeomorphisms in Σ, and the time development of the kl variables gij (x), π (y). One can ask what effect the constraints (3) would have when applied on wave functions ψ(g). If the brackets (5) are represented by letting the momenta πij act by differentiation δ πij (x) (6) ∼ δgij (x) a This is standard ADM notation: gij and R refer to the induced metric and scalar (4) curvature of a time slice, while the spacetime metric and curvature are denoted gµν and (4)R. October 24, 2018 9:11 Proceedings Trim Size: 9in x 6in des 3 and the metric components gij (x) by multiplication, the supermomentum constraint ψ(g) = 0 is easy to interpret, since Hi 3 i δΨ(g) d x N i Ψ(g)= δΨ(g)= ( iNj + j Ni) (7) Z H δg (x) ∇ ∇ Σ ij and implies that one should identify wave functions of metrics gij andg ˜ij when they differ as g˜ = g + N + N . (8) ij ij ∇i j ∇j i But (8) is a Lie derivative, or coordinate transformation, in the spatial surface Σ, so the supermomentum constraint reflects the freedom to choose the 3 spatial coordinates on Σ. The space of metrics (8) withg ˜ij identified 4 with gij was named superspace in 1963 by Wheeler . The super-Hamiltonian constraint ψ(g) = 0 (also known as the H Wheeler DeWitt equation 5) is much harder to interpret, and alone does not generate the dynamics, or time reparametrization invariance, of wave functions ψ(g). Instead one needs to use the full Hamiltonian, namely the combination 3 i d x N i + N . (9) Z H H Σ The gravitational field in 4 spacetime dimensions has correctly (for a mass- less field) 2 independent degrees of freedom per spacetime point. This is most easily seen by noting that the induced metric of a time slice gij has 6 independent components, and there are the 4 constraints (3). 2. 2 + 1–Dimensional ADM Decomposition In 2 + 1 spacetime dimensions the description of Section (1) is essentially 1 identical, apart from a factor of 2 in the super-Hamiltonian (3). The count- ing of degrees of freedom is, however, quite different. There are in fact zero degrees of freedom, and this can be seen in several ways. The simplest is perhaps to note that now the induced matric gij ,i,j =1, 2 has only 3 inde- pendent components, but there are 3 constraints =0, = 0 analogous H Hi to (3). Alternatively, since the Weyl tensor vanishes in 3 dimensions (but 6 not in 4, see ), it follows that the full Riemann curvature tensor Rαβµν can be decomposed uniquely in terms of only the Ricci tensor Rµν the scalar curvature R and the metric tensor gµν itself. R = g R g R g R + g R λµνk λν µk − µν λk − λk µν µk λν 1 + R(g g g g ) (10) 2 µν λk − λν µk October 24, 2018 9:11 Proceedings Trim Size: 9in x 6in des 4 d2(d2 1) In fact in d dimensions Rαβµν has 12− independent degrees of freedom d(d+1) and Rµν has 2 . These coincide when d = 3. In terms of the Einstein tensor Gαβ (equation (4)) the decomposition (10) is αβ Rλµνk = ǫλµβ ǫνkα G (11) so that when Einstein’s vacuum equations Gαβ = 0 are satisfied, the full curvature tensor (all components) are zero, i.e. Rλµνk = 0 and spacetime is flat. Thus vacuum solutions of Einstein’s equations correspond to flat spacetimes, and there are no local degrees of freedom. It is possible, however, to solve the field equations and introduce some dynamics, in several ways. The first - developed extensively by Deser et al 7 and others, is to add sources, or matter, thus creating local degrees of freedom. When Einstein’s equations read Gαβ = T αβ (12) where T αβ is the stress-energy tensor of the sources, the curvature (11) is no longer zero, but is proportional, from (12), to T αβ. The second creates propagating massive gravitational modes by adding a topological term to the action, always possible in an odd number of di- mensions 6. For gravity in 3 dimensions, this is the Chern-Simons form ab 2 ac d (ω dωab + ω ω ωda) (13) Z ∧ 3 ∧ c ∧ ab where the components of the spin connection ωµ are to be considered as a functionals of the triads eµ by solving the torsion equation. Ra = dea ωab e =0 − ∧ b a b with eµeν ηab = gµν . Variation of (13) with respect to the metric tensor gµν gives the Cotton tensor µν 1 µλβ ν 1 ν C = g− 2 ǫ D R δ R λ β − 4 β which is symmetric, traceless, conserved, and vanishes if the theory is con- formally invariant. Therefore, adding the Chern-Simons term (13) to the 1 three-dimensional scalar curvature action (1) with a constant factor µ leads to the field equations 1 Gµν + Cµν =0 µ October 24, 2018 9:11 Proceedings Trim Size: 9in x 6in des 5 which can be transformed into 2 2 + µ Rµν = terms in Rµν (14) In the linearized limit the R.H.S. of (14) vanishes and it is shown in 6 that the solutions of (14) correspond to massive, spin 2, particles. ± A way to introduce global rather than local degrees of freedom in flat spacetime is to consider non trivial topologies. Recall that curvature is defined by commutators of covariant derivatives, or, by parallel transport around non-collapsible curves i.e. curves which are not homotopic to the identity. The change effected by parallel transport around closed curves of this type is often called holonomy - and is used to characterise flat space- times.